Em05-A Excitation Transfer, Subsystem Purity, and Referenced Information Recovery Exact Diagnostics and Population-Preserving Noise in a Fermionic Extraction Channel

We analyze a completely specified even-parity excitation-qubit extraction channel and its full complementary channel. The signed isometry yields exact formulas for transferred occupation, occupation-basis coherence, subsystem purity, reference correlations, and deterministic recovery performance. For a Bell input, we determine the accessible and residual spectra, their mutual-information balance, and the reference–accessible negativity. A complete specified decoder has half-diamond error $1-\eta$, Bell fidelity $(1+3\eta )/4$, worst-case target overlap $\eta$, and decoded reference concurrence $\eta$. We distinguish these formulas from optimal recovery and determine the optimal errors at zero and full transfer.

An independently declared population-preserving output dephasing experiment retains its additional environment. At full transfer, its optimal recovery error is $(1-\lambda )/2$, while the Bell-input accessible marginal, occupation, and purity are independent of $\lambda$. Complete dephasing therefore leaves perfect occupation delivery and an unchanged accessible marginal, but destroys reference entanglement and prevents exact quantum recovery. These results provide a self-contained information-flow benchmark. They neither identify the channel with black-hole evaporation nor establish a gravitational Page curve. No numerical execution or completed verification package is asserted.

The following quantities are often considered together:

1. excitation transferred to an accessible subsystem;

2. coherence in a specified occupation basis;

3. entanglement across a specified partition;

4. purity of an accessible or residual marginal;

5. recoverability of an unknown quantum message, including its correlations with an external reference.

They answer different questions.

This manuscript constructs an exact comparison within a specified channel family. In particular, we prove that a pure accessible subsystem can contain no message information, and that a mixed accessible subsystem can carry a perfectly recoverable quantum message. We also construct two experiments with identical Bell-input accessible marginals and identical excitation populations, but different optimal quantum-recovery errors.

Finite-dimensional evaporation models have substantial precedent. Avery studies a class encompassing unitary and nonunitary toy evaporation processes Avery. Hayden and Preskill formulate an information-retrieval task under assumptions including unitary, rapidly mixing internal dynamics and extensive access to radiation HaydenPreskill. Neither model is identified with the extractor specified below.

Reference-based exact recovery is part of the established quantum-error-correction framework KnillLaflamme. Channel continuity and information–disturbance relations are developed by Kretschmann, Schlingemann, and Werner KSW. Bény and Oreshkov formulate approximate-correctability conditions and an environmental dual optimization using worst-case entanglement fidelity BenyOreshkov.

Our contribution is the explicit diagnostic comparison, complete signed channel construction, and population-preserving noise counterexample for the declared extraction family. We do not claim a new general information–disturbance principle, a new definition of quantum recovery, or first-in-literature priority for the elementary dephasing and replacement-channel calculations.

The finite channel is restated and proved independently here. The supplied Em03-H-v8 and Em04-A-v2 manuscripts motivate the comparison, but are not formal premises of its proofs. The XY-chain diagnostic identities of Em04-A are not imported into this different excitation code.

Ordinary quantum states, channels, tensor products, and Born probabilities are assumed. The mathematical results are analytical. No compiler log, numerical archive, executed symbolic check, machine-checked proof, or physical receiver accompanies this source.

The parameter $\eta$ is an extraction probability. It is not identified with black-hole age, radiated entropy, or evaporation time. No passive evaporation, horizon state, gravitational factorization, backreaction estimate, or microscopic gravity dictionary is supplied.

For a finite matrix $X$, write

For states,

For channels with the same input and output spaces,

For a difference of finite-dimensional channels, this equals the supremum of the output trace distance over referenced input states. A reference of input dimension suffices. These are standard finite-channel conventions Watrous.

Let $M$ be a logical qubit, $Q$ an external reference, $R$ the accessible two-mode output, and $E$ the residual two-mode output. The decoded qubit is $M^{\prime }$. All logarithms in entropy and entanglement expressions are base two.

For a qubit-output channel $\Lambda$, define

For a purification $|{\psi }_{\rho }{⟩}_{QM}$ of ${\rho }_M$, define

The target overlaps are squared fidelities against their pure targets.

For an accessible channel $\mathcal{N}:M\to R$, set

where the input and decoded logical bases are identified.

A replacement channel is

and ${\mathrm{Ad}}_U(X)=UX{U}^{†}$.

A CPTP map contracts trace norm on Hermitian inputs. Consequently trace distance contracts under channels, and channel half-diamond distance contracts under fixed CPTP postprocessing. For states $\rho ,\sigma$ and an effect $0\le F\le I$,

Proof. For the Hermitian Jordan decomposition $X=X_{+}-X_{-}$, positivity and trace preservation give

The same argument applies with a reference because a CPTP map remains positive after tensoring with its identity. For the effect bound, the trace-zero difference $\rho -\sigma$ has positive and negative parts of equal trace ${\Vert \rho -\sigma \Vert }_1/2$. Testing either part against $F$ proves the assertion.

Set

with ordered occupation bases

The two occupations in each subsystem refer to two distinct species or modes. The fixed tensor representation is bath first, residual second.

Let

Here $t$ is a residual probability, not a time variable.

The map $V_{\eta }:{\mathbb{C}}^2\to {\mathcal{H}}_R\otimes {\mathcal{H}}_E$ is

Define the accessible and complete complementary channels by

The minus sign in (eq:Vone) is the bath-first CAR sign. For residual creators ${e_1}^{†},{e_2}^{†}$ and bath creators ${b_1}^{†},{b_2}^{†}$, expand

Moving bath creators before residual creators gives precisely (eq:Vone). Both logical basis vectors have even total fermionic parity. The reference $Q$ is an abstract logical register, not an additional occupation mode. All partial traces use the specified occupation representation.

If an uncorrected amplitude is $q=\sqrt{\eta }{e}^{\mathrm{i}\alpha }$, the occupied column has coefficients $q^2,q\sqrt{t},-q\sqrt{t},t$. Applying the accessible phase correction $e^{-\mathrm{i}\alpha N_R}$ gives Definition def:extractor. The phase is fixed independently of the logical input. The ideal-phase results below do not certify an estimated or implemented phase correction.

The map $V_{\eta }$ is an isometry for every $\eta \in [0,1]$. Both channels in Definition def:extractor are CPTP.

A Kraus family for ${\mathcal{N}}_{\eta }$, indexed by residual occupation, is

The complete complement is

These formulas retain arbitrary external references.

Proof. The two columns of $V_{\eta }$ are orthogonal, and the occupied column has squared norm

Thus ${V_{\eta }}^{†}{V}_{\eta }=I_2$. Reading its coefficients in the residual basis gives (eq:kraus). Direct calculation gives

The accessible map is therefore CPTP. The second partial trace of the same isometry is CPTP and gives (eq:complement). Tensoring the defining isometry with $I_Q$ proves the reference statement.

Tensoring $V_{\eta }$ with a fixed pure spectator state does not change any formula below. The spectator remains part of the complete environment, but adds no entropy or logical dependence. An input-correlated or mixed spectator is not covered by this simplification without retaining its preparation purification.

Write an arbitrary logical density matrix as

In the declared occupation bases,

\rho_R=\mathcal N_\eta(\rho_M)
 &=
 \begin{pmatrix}
 1-p+pt^2&0&0&\eta z\\
 0&p\eta t&0&0\\
 0&0&p\eta t&0\\
 \eta z^*&0&0&p\eta^2
 \end{pmatrix},
 \label{eq:rhoR}\\
 \rho_E=\mathcal C_\eta(\rho_M)
 &=
 \begin{pmatrix}
 1-p+p\eta^2&0&0&t z\\
 0&p\eta t&0&0\\
 0&0&p\eta t&0\\
 t z^*&0&0&pt^2
 \end{pmatrix}.
 \label{eq:rhoE}

For

on the respective subsystems,

The distinguished occupation-basis coherences are

The accessible purity is

The residual purity follows by interchanging $\eta$ and $t$.

Proof. Insert (eq:input) into the Kraus expressions. Only $A_{00}$ contributes the accessible coherence. For the complement, the only off-diagonal residual matrix units are $00,11$ and their adjoints, with coefficient $tz$. This gives (eq:rhoR) and (eq:rhoE).

Tracing either marginal against its number operator gives

The coherence formulas are their displayed entries. For a Hermitian matrix, $\mathrm{Tr}{\rho }^2={\sum }_{i,j}|{\rho }_{ij}{|}^2$. Applying this identity to (eq:rhoR) gives (eq:purity).

For every logical state,

The nonzero eigenvalues of $V_{\eta }{\rho }_M{V}_{\eta }^{†}$ are those of ${\rho }_M$. In particular,

For every purified input, the joint referenced output is pure. Thus the declared isometry does not globally destroy information.

Proof. The occupation statement follows from $\eta +t=1$. An isometry preserves inner products and embeds the spectral decomposition of its input with the same nonzero eigenvalues. Tensoring with a reference preserves the norm and purity of a purified input.

Occupation means depend only on $p$. The purity also depends on $|z{|}^2$, and the coherence depends on the declared basis. A single input's marginal purity or coherence is not a criterion for recovery of all referenced inputs. Loss of accessible information means failure of recovery from $R$, not nonunitarity of the retained $RE$ experiment.

Set

The logical marginal is $I_M/2$, so $p=1/2$ and $z=0$ in the marginal formulas.

For the Bell input,

including zero multiplicities. The residual spectrum is obtained by interchanging $\eta$ and $t$.

With

using $0{\log }_{2}0=0$, one has

Consequently,

and

Proof. For the Bell input the matrices (eq:rhoR) and (eq:rhoE) are diagonal, giving the spectra. The global state $|{\Psi }_{\eta }⟩$ is pure and $S(Q)=1$. For a pure bipartite state the two reduced states have the same nonzero spectrum: expand its coefficient matrix and compare the nonzero eigenvalues of $A{A}^{†}$ and $A^{†}A$. Apply this to $QR|E$ and $QE|R$ to obtain $S(QR)=S(E)$ and $S(QE)=S(R)$. This proves (eq:MIr)–(eq:balance).

At the endpoints the spectra give entropies $0,1$ in opposite orders. At $\eta =t=1/2$, the spectra coincide. Squaring and summing the accessible eigenvalues gives the three displayed purities.

Equation (eq:balance) is a pure-state mutual-information identity. Mutual information includes classical correlations. It is neither a statement that both subsystems contain an independently recoverable qubit nor a gravitational entropy law.

For an ordinary bipartite density matrix, define negativity by

with partial transpose in the declared reference basis VidalWerner.

For the Bell-input state,

Thus the reference and accessible output are entangled for every $\eta >0$.

Proof. The only coherence of ${\rho }_{QR}$ connects $|0{⟩}_Q|00{⟩}_R$ and $|1{⟩}_Q|11{⟩}_R$, with coefficient $\eta /2$. Under $T_Q$, it enters the block on $|0{⟩}_Q|11{⟩}_R,|1{⟩}_Q|00{⟩}_R$:

Its eigenvalues are

All remaining eigenvalues are nonnegative diagonal entries or zeros. The negative eigenvalue therefore gives (eq:rawneg). A separable state's partial transpose is positive, because transposition preserves positivity of each factor in a separable decomposition. The strictly negative eigenvalue at $\eta >0$ proves entanglement.

This proposition concerns $Q|R$, not the partition between the two bath modes. Reference entanglement can survive even when exact recovery is impossible.

Let

Define

Every off-code event is retained and mapped to the maximally mixed decoded qubit.

The map ${\mathcal{D}}_{\star }$ is CPTP on every input state on ${\mathcal{H}}_R$.

Proof. Use the retained Kraus operator $W^{†}$, together with

Their effects sum to $P+|01⟩⟨01|+|10⟩⟨10|=I_R$. Their Kraus sum is exactly (eq:decoder).

The decoded channel is

It satisfies

For the Bell-input decoded state,

the two-qubit concurrence is

No optimality of ${\mathcal{D}}_{\star }$ is asserted for intermediate $\eta$.

Proof. The odd occupations in (eq:rhoR) contribute $p\eta t{I}_2$ after decoding. The decoded lower-right entry is $p{\eta }^2+p\eta t=p\eta$, and its off-diagonal entry is $\eta z$. The upper-left entry is $1-p\eta$. This proves (eq:decoded).

A half-diamond distance between channels is at most one. The identity and replacement by $|0⟩$ attain distance one on input $|1⟩$. Homogeneity therefore gives (eq:specifiedError).

The replacement term's Bell overlap is $1/4$, proving (eq:BellFid). For any purified input, the target overlap of (eq:decoded) is at least $\eta$, because the replacement contribution is nonnegative. Input $|1⟩$ gives zero replacement contribution and attains $\eta$, proving (eq:WorstFid).

For concurrence, use Wootters' definition Wootters: the concurrence is the positive part of the largest square-root eigenvalue of $\omega \tilde{\omega }$ minus the other three, where

In the ordered basis $00,01,10,11$, (eq:decodedBell) is

The square-root eigenvalues are

Their difference is $\eta$.

For $0\le \eta <1$,

For all $\eta$,

At the endpoints,

In particular, the specified decoder's error one at $\eta =0$ is not the optimal error.

Proof. Let ${\rho }_i={\mathcal{N}}_{\eta }(|i⟩⟨i|)$. These states are diagonal and

For any decoder with error $\epsilon$, its decoded basis inputs are within $\epsilon$ of their respective orthogonal targets. The triangle inequality and contraction imply

Thus $\epsilon \ge t^2/2$.

The specified decoder gives the upper bound $1-\eta$. Another decoder ignores its input and returns $I_2/2$. For Pauli matrices $X,Y,Z$,

The triangle inequality gives its distance to the identity at most $3/4$.

At $\eta =0$, every decoded channel is a replacement ${\mathcal{R}}_{\tau }$. Its Bell output is $I_Q/2\otimes \tau$, whose overlap with $|\Phi ⟩$ is $1/4$, independently of $\tau$. The Bell-projector effect and Lemma lem:contraction give distance at least $3/4$. The maximally mixed replacement attains the preceding upper bound. At $\eta =1$, ${\mathcal{N}}_1={\mathrm{Ad}}_W$, and the specified decoder is its left inverse.

At $\eta =0$, the accessible subsystem is pure vacuum for every message, but the optimal recovery error is $3/4$. At $\eta =1$, a Bell input produces an accessible marginal $W(I_2/2)W^{†}$ of purity $1/2$, but recovery is exact.

At full transfer the residual subsystem is pure vacuum. That purity certifies neither residual recovery nor global loss: the message has moved to $R$.

The original extractor is now followed by a separately specified noise channel. It is a different experiment, not a hidden property of Definition def:extractor.

On the entire accessible space define

It is Hermitian, unitary, parity preserving, and commutes with every occupation projector and with $N_R$.

For $0\le \lambda \le 1$, put

and define

Let $F={\mathbb{C}}^2$ be an additional environmental register. The isometry

is a dilation of ${{\mathcal{Z}}^{\mathcal{R}}}_{\lambda }$.

Define

with the fixed output reordering $RFE\to REF$. The new accessible channel is

Its complete environment is $EF$, not merely $E$.

A Kraus family for ${\mathcal{N}}_{\eta ,\lambda }$ is

Its complete complement is

The marginal ${\mathcal{N}}_{\eta ,\lambda }({\rho }_M)$ is (eq:rhoR) with $\eta z$ replaced by $\lambda \eta z$, and its adjoint entry replaced accordingly. All occupation probabilities and $⟨N_R⟩$ are unchanged for every input.

Proof. Since $a_{\lambda }+b_{\lambda }=1$ and ${S_R}^2=I$, the Kraus effects sum to $I_M$. Taking both partial traces of ${\tilde{V}}_{\eta ,\lambda }$ gives the channels and complement. Conjugation by $S_R$ leaves diagonal entries unchanged and negates the $00,11$ coherence. The weighted sum multiplies that coherence by $a_{\lambda }-b_{\lambda }=\lambda$.

At $\lambda =0$, the noisy accessible channel has no dependence on $z$:

It is a measure-and-prepare channel. Its referenced output is separable, because it is a sum of positive reference operators tensored with prepared output states. Complete dephasing therefore destroys accessible reference entanglement for every $\eta$.

At $\eta =1$, set

Their overlap is $⟨u_{\lambda }|v_{\lambda }⟩=\lambda$.

The full-transfer noisy isometry is

For the full-transfer dephasing experiment,

The specified decoder attains this optimum. For its Bell-input decoded output,

The Bell-input accessible marginal is independent of $\lambda$:

Writing

with its continuous endpoint values, the mutual informations are

At complete dephasing, $\lambda =0$, occupation delivery is perfect, decoded concurrence is zero, and the optimal recovery error is $1/2$.

Proof. On the logical qubit, the specified decoded channel is

The channels ${\mathrm{Ad}}_Z$ and $\mathrm{id}$ have half-diamond distance one: input $|+⟩$ gives orthogonal outputs. Hence

For the lower bound valid for every decoder, use logical inputs $|+⟩$ and $|-⟩$. Their noisy accessible outputs have trace distance $\lambda$, whereas their ideal targets have distance one. A decoder of error $\epsilon$ must therefore satisfy

Thus $\epsilon \ge (1-\lambda )/2$, proving optimality.

The decoded Bell state is

Its overlap with $|\Phi ⟩$ is $(1+\lambda )/2$. Its spin-flip square-root eigenvalues are $(1+\lambda )/2,(1-\lambda )/2,0,0$, so its concurrence is $\lambda$.

Tracing the reference in (eq:fullNoisy) gives (eq:identicalMarginal). The $F$ marginal is

Thus $S(EF)=h_2(a_{\lambda })$, while $S(Q)=S(R)=1$. The joint state $QREF$ is pure, so the same pure-state entropy argument as in Theorem thm:information gives (eq:noiseMIr) and (eq:noiseMIe).

For the Bell input, the noiseless full-transfer experiment $(\eta ,\lambda )=(1,1)$ and the completely dephased experiment $(1,0)$ have identical accessible density matrices. They therefore have identical accessible occupation probabilities, purity, and every other observable depending only on that marginal.

Nevertheless their optimal recovery errors are respectively $0$ and $1/2$, their decoded reference concurrences are $1$ and $0$, and their reference–accessible mutual informations are $2$ and $1$.

Proof. Apply Theorem thm:fullNoise at its two endpoints.

At $\lambda =0$, $u_0$ and $v_0$ are orthogonal. For the Bell input, the complete output has the form

The accessible reference state is only classically correlated. The retained $RF$ isometry still preserves the input qubit. Discarding $F$ creates the accessible dephasing channel. Keeping only the original vacuum residual $E$ would incorrectly hide the environmental information.

The preceding formulas are exact identities for declared channels. They do not assert a validated amplitude for a particular numerical or continuum experiment.

If a validated calculation establishes

then the ideal-phase specified decoder satisfies

For a fixed known $p$,

Purity can be enclosed by outward evaluation of (eq:purity), but it must not be presumed monotone in $\eta$.

Proof. The first four quantities are affine in $\eta$ with the displayed monotonicity. The purity is an explicit polynomial in $\eta ,t,p,|z{|}^2$, so inclusion-preserving interval evaluation encloses its value.

Suppose two accessible channels with the same types obey

For any fixed CPTP decoder $\mathcal{D}$,

Their optimal recovery errors also satisfy

For every common referenced input, their output states have trace distance at most $\xi$. For the corresponding accessible marginals,

and

For a fixed decoder, the Bell fidelities differ by at most $\xi$.

Proof. Postprocessing contraction and the reverse triangle inequality prove the fixed-decoder statement. Apply it for every decoder and take infima in both directions to obtain optimal-error stability. The referenced state bound is the definition of channel distance.

For purity,

Since the state difference has trace zero, center the occupation operator:

because ${\Vert N_R-I\Vert }_{\infty }=1$. For distinct basis vectors $u,v$ and Hermitian $X$,

This follows by testing against a phase-aligned Hermitian off-diagonal operator of norm one. It gives the coherence-entry estimate. The Bell-projector effect bound gives the fidelity estimate.

A transmission interval does not automatically certify:

• a physical gravitational identification;

• the implementation of the decoder;

• the accuracy of a phase correction;

• a common identification of residual wavefunctions across different experiments;

• a comparison of their physical complementary channels.

State residual bounds of the kind developed in Em04-A are useful interfaces, but a single-state bound is not automatically a referenced channel bound. A channel certificate must control the specified channel on arbitrary referenced inputs.

For an implemented phase error $\delta$, the noiseless decoded channel instead has the form

Its half-diamond error is at most

Indeed the relevant unitary channel has distance $|\sin \delta |$ from the identity: its numerical range has minimal modulus $|\cos \delta |$, and the pure referenced output distance is the square root of one minus the squared overlap. This phase allowance and any additional implementation error must be charged rather than absorbed into an unverified ideal-phase statement.

{}1.15

For the declared extraction and output-noise experiments:

1. The signed extraction map and its complete complement are physical for every $\eta \in [0,1]$.

2. Occupation, coherence, purity, Bell-input spectra, reference mutual information, and reference negativity obey the proved identities.

3. The specified decoder has error $1-\eta$, Bell fidelity $(1+3\eta )/4$, worst-case target overlap $\eta$, and decoded concurrence $\eta$.

4. Exact recovery is impossible at $\eta <1$, but the specified decoder is not presumed optimal there. Optimal recovery errors at $\eta =0,1$ are $3/4,0$.

5. Population-preserving dephasing retains an additional complete environment. At full transfer its optimal recovery error is $(1-\lambda )/2$, although the Bell-input accessible marginal is independent of $\lambda$.

6. Accessible marginal purity, occupation delivery, and even knowledge of the complete accessible marginal for a single Bell-input experiment do not replace a referenced channel-recovery criterion.

These are finite-channel information-flow results. They do not establish black-hole evaporation, a gravitational Page curve, or a solution to the black-hole information paradox.

Proof. Theorem thm:channels proves (i). Theorem thm:marginals, Theorem thm:information, and Proposition prop:rawneg prove (ii). Theorem thm:specified proves (iii). Proposition prop:optimal proves (iv). Proposition prop:noisy and Theorem thm:fullNoise prove (v). The purity endpoints and Corollary cor:separation prove (vi). None of the hypotheses specifies a gravitational identification.

The natural application is an exactly solvable benchmark for information-flow diagnostics and recovery certification. A sequential evaporation-like model would require newly specified fresh output registers, an access schedule, internal dynamics, and a complete environment at every step. Such a construction is not obtained by relabeling $\eta$ as time.

The finite family includes $\eta =1$ as an exact mathematical endpoint. It does not imply that the retained-storage continuum experiment of Em03-H attains that endpoint.

For clarity, the obstruction can be restated independently. Let

Take $t_0\ge 0$, a initially vacant bath level, and the initial one-species state

For finite extraction time $T$, use bounded piecewise continuous real controls $v(u),\kappa (u)$ and the generator

on $\mathfrak{h}\oplus \mathbb{C}$.

Relative to $h\oplus 1$, the perturbation is bounded, self-adjoint, and finite rank. Its interaction-picture Dyson series gives a unitary propagator. Put

Variation of constants gives

In this declared continuum model, $|B(T)|<1$ for finite $T$.

Proof. If $|B(T)|=1$, unitarity forces $\psi (T)=0$. Since $f(\omega )>0$, the packet formula implies

for almost every positive $\omega$. The function $c$ is bounded and integrable. Both sides extend to entire functions. Their equality on a full-measure subset of a real interval gives equality everywhere by continuity and the identity theorem.

At $\omega =\mathrm{i}y$, $y>0$, the left side is $e^{y{t}_0}$, whereas

The right side tends to zero and the left side is at least one, a contradiction.

This proposition supplies neither a uniform positive numerical error floor nor a quarter-error exclusion. It distinguishes physical endpoint attainability in this model from endpoint identities of the abstract channel family. No numerical value of $B(T)$ is asserted here.

The construction incorporates the following corrections to the motivating proposal:

1. The transfer family is stated independently of its continuum realization. Full transfer is not falsely reported as a finite-time continuum result.

2. The specified decoder's zero-transfer error is one, whereas the optimal deterministic error is $3/4$.

3. The Bell-input spectra and mutual-information values are labeled as input-specific.

4. The complete noisy complement includes the dephasing register $F$.

5. At full transfer, complete dephasing has optimal recovery error $1/2$, proved for all deterministic decoders.

6. Reference concurrence is calculated using the spin-flip spectrum, not identified with a bath-mode entanglement measure.

7. The extraction parameter is not assumed monotone along a physical time trajectory.

8. The Knill–Laflamme journal citation uses the publisher's Physical Review A article, rather than the conflicting journal metadata displayed on the supplied arXiv landing page.

Before claiming an executed verification package, independently:

• compile until all references stabilize and inspect the PDF;

• construct the isometry matrices directly and check ${V_{\eta }}^{†}{V}_{\eta }=I$;

• trace the full referenced states independently of the displayed marginal formulas;

• compare spin-flip and partial-transpose spectra with the analytical expressions, including $\eta =0,1$;

• verify the noisy isometry with all $EF$ registers retained;

• retain input data, package versions, tolerances, raw outputs, and failures for every executed numerical test.

These are proposed checks, not execution claims. Floating-point agreement is not a rigorous certificate.

. This single source contains the model, proofs, and internal bibliography. No external data are required for typesetting.

. No compilation, numerical calculation, independent symbolic execution, or formal proof checking is claimed.

interpretation. The work is a finite-channel benchmark motivated by information-flow questions. It does not identify the residual subsystem with a black-hole interior, the accessible bath with Hawking radiation, or either parameter with gravitational evolution.

. Quantum recovery, complementary channels, entropy identities, concurrence, negativity, and dephasing are established frameworks. The model-specific calculations are proved here without claiming a new general foundation or exhaustive novelty verification.

Avery

S. G. Avery, “Qubit Models of Black Hole Evaporation,” arXiv:1109.2911v2 [hep-th] (2012). Used as context for finite-dimensional evaporation models, not as a derivation of the present extractor.

HaydenPreskill

P. Hayden and J. Preskill, “Black holes as mirrors: quantum information in random subsystems,” Journal of High Energy Physics 09 (2007), 120. DOI: 10.1088/1126-6708/2007/09/120. arXiv:0708.4025. Used as recovery-task precedent under its own dynamical and access assumptions.

KnillLaflamme

E. Knill and R. Laflamme, “Theory of quantum error-correcting codes,” Physical Review A 55, 900–911 (1997). DOI: 10.1103/PhysRevA.55.900. Preprint title: “A Theory of Quantum Error-Correcting Codes,” arXiv:quant-ph/9604034. Used for the established reference-based exact-recovery framework.

KSW

D. Kretschmann, D. Schlingemann, and R. F. Werner, “The Information-Disturbance Tradeoff and the Continuity of Stinespring's Representation,” IEEE Transactions on Information Theory 54, 1708–1717 (2008). DOI: 10.1109/TIT.2008.917696. arXiv:quant-ph/0605009. Used as information–disturbance and channel-continuity background.

BenyOreshkov

C. Bény and O. Oreshkov, “General conditions for approximate quantum error correction and near-optimal recovery channels,” Physical Review Letters 104, 120501 (2010). DOI: 10.1103/PhysRevLett.104.120501. arXiv:0907.5391. Used as approximate-recovery background; its fidelity metric is not silently identified with the half-diamond metric used here.

Watrous

J. Watrous, The Theory of Quantum Information, Cambridge University Press (2018). Used for finite channels, referenced channel norms, and entropy conventions.

Wootters

W. K. Wootters, “Entanglement of formation of an arbitrary state of two qubits,” Physical Review Letters 80, 2245–2248 (1998). DOI: 10.1103/PhysRevLett.80.2245. arXiv:quant-ph/9709029. Used for the concurrence definition.

VidalWerner

G. Vidal and R. F. Werner, “Computable measure of entanglement,” Physical Review A 65, 032314 (2002). DOI: 10.1103/PhysRevA.65.032314. arXiv:quant-ph/0102117. Used for the negativity definition.